On functions of bounded n-th variation
Abstract
The class of functions of bounded n-th variation, denoted by BVn[a,b], was introduced by M.T. Popoviciu in 1933. In 1979 A.M. Russell in [6] proved the Jordan-type decomposition theorem for functions from this class, then, applying this result, showed that each BVn[a,b], with a suitable norm, is a Banach space. For n=0 and n=1 the above facts are well known classical results (cf., for instance, [5], [7]).
However, the proofs given by A . M . Russell for n≥2, based on some properties of divided differences, are rather complicated. The aim of this note, is to give an essentially simpler arguments both, for the Jordan-type decomposition theorem as well as for the completness of the space BVn[a,b]. In our proofs we apply the Popoviciu theorem and the fact that for every positive integer n≥2, a function f is n-convex iff the derivative f' is (n-1)-convex.
References
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3. M. Kuczma, An introduction to the theory of functional equations and inequalities, PWN, Warszawa-Kraków-Katowice 1985.
4. M.T. Popoviciu, Sur quelques proprietes des functions d'une variable reele convexes d'orde superieur, Matheinatica, Cluj 8.
5. A.W. Roberts, D.E. Varberg, Functions of bounded convexity, Bulletin of AMS 75.3, (1969), 568-572.
6. A.M. Russell, A commutative Banach algebra of functions of generalized variation, Pac. J. Math. 84, no 2 (1979), 455-463.
7. A.E. Taylor, Introduction to functional analysis, Wiley, New York 1967.
8. D.E. Varberg, Convex functions, Academic Press 1973.
Instytut Matematyki i Informatyki, WSP w Częstochowie Poland
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